a hybrid meshless/spectral-element method for the … · a hybrid meshless/spectral-element method...

A Hybrid Meshless/Spectral-Element Methodfor the Shallow Water Equations on the Sphere

Christopher D. Blakely

Center for Scientific Computation and Mathmematical ModelingUniversity of Maryland, College Park [email protected],

www.cscamm.umd.edu/~cblakely

1 Introduction and Motivations

The purpose of this paper will focus on constructing an innovative hybridapproximation method for geophysical fluid dynamics. To accomplish sucha task, we will focus on the shallow-water equations which provide a usefulmodel to global climate modeling because their solutions include nonlineareffects and wave structures similar to those of the full primitive equations ofthe atmosphere. The main backbone of this hybrid meshless/spectral-elementshallow-water model will be focusing on incorporating a unique regional scaleapproximation method. This regional scale method will be accomplished byusing a robust meshless approximation scheme called the empirical Backus-Gilbert reproducing kernel developed by C. Blakely in [5]. The advantage ofsuch a hybrid approximation is two-fold: 1) High-order approximation resultscan be obtained in complex shaped geometries without the need of a mesh.Thus, no remeshing of a local region into smaller rectangles is needed, ulti-mately speeding up the computation time; 2) The Backus-Gilbert reproducingkernel method has been shown to be endowed with the unique power of ignor-ing oscillatory effects in scattered data. This will be useful when the spectralapproximation forms high oscillations due to discontinuities in the data.

This article is organized as follows. Section (2) begins with a brief re-view of the shallow-water equations defined on the cubed-sphere and it’s dis-cretization in time using a semi-implicit time stepping scheme. We followthis discussion in section (3) with a brief review of the spectral-element andBackus-Gilbert reproducing kernel discretization methods of the time-discreteshallow-water equations. Next, we present the three-field variational formu-lation which effectively couples the two types of approximations in a weaksense by introducing two additional approximation spaces on the interfacesbetween the approximation, akin to domain decompositon and the mortar el-ement method. Implementation of the coupling is then given and then finally,in order to verify the mathematical correctness of the algorithms presentedin this paper and to validate the performance hybrid model, we conclude the

2 Christopher D. Blakely

paper with some standardized test cases which were proposed by Williamsonet al. in [22] (W92).

2 The shallow-water equations on the cubed-sphere

Being the simplest form of motion equations that can approximate the hor-izontal structure of the atmosphere or the dynamics of oceans, the shallow-water equations have been used as a robust testing model in atmospheric andoceanic sciences. The solutions can represent certain types of motion includ-ing Rossby waves and inertia-gravity waves while describing an incompressiblefluid subject to gravitational and rotating acceleration. The governing equa-tions for the inviscid flow of a thin layer of fluid in 2-D are the horizontalmomentum and continuity equations for the velocity u = (u1, u2) and thegeopotential height η .

While there are many different ways of defining the shallow-water equa-tions, we focus in this model on cubed-sphere geometry originally proposedby Sadourny in [17] and used in other global models in recent years such as[20] and [21]. We begin by a brief review of the cubed-sphere while adoptingnotational conventions from [20]. Consider a cube inscribed inside a spherewhere each corner of the cube is a pointin the sphere and where each face ofthe cube is subdivided into NE subregions. The goal is to project each faceof the cube onto the sphere and in effect, obtain a quasi-uniform sphericalgrid of 6×NE subregions which can be further subdivided into many spectralelement and meshless collocation subregions. In the mapping of the cube tosphere, each face of the cube is constructed with a local coordinate systemand employs metric terms for transforming between the cube and the spherewhich will now be defined.

Let (α, β) be equal angular coordinates such that −π/4 ≤ α, β ≤ π/4.Then any x1 and x2 on a face Pi of the cube is related through x1 = tanα, x2 =tanβ. We denote r the corresponding position vector on the sphere with lon-gitude λ and latitude θ. For such an equiangular projection, we define basisvectors a1 = rα and a2 = rβ which may be written as

rα =1

cos2 αrx1

, rβ =1

cos2 βrx2

(1)

where rx1and rx2

are defined as rx1=(

cos θ λx1, θx1

)and rx2

=(

cos θ λx2, θx2

)

The metric tensor gij , i, j ∈ [1, 2] can be derived as

gij = ai · aj =1

r4 cos2 α cos2 β

[1 + tan2 α − tanα tanβ− tanα tanβ 1 + tan2 β

],

where r2 = 1 + tan2 α + tan2 β and the Jacobian of the transformation andthe matrix A are, respectively,

Title Suppressed Due to Excessive Length 3

√g = [det(gij)]

1/2 =1

r3 cos2 α cos2 β, A =

cos θ λα cos θ λα

θα θβ

.

While using the definition of gij given in (2), we can write transformationsbetween covariant and contravariant components of a vector v as

u2

u2

=

g11 g12

g21 g22

u1

u2

,

u1

u2

=

g11 g12

g21 g22

u1

u2

. (2)

With the metric terms defined, we can now write the shallow water equa-tions in in the curvilinear coordinates system to be integrated on the cubed-sphere. In such a coordinate system, the shallow-water equations can be writ-ten as follows

∂ui

∂t= −gij

[εjku

kg(f + ζ) +∂

∂xj

(1

2uku

k)

+∂η

∂xj

],

∂η′

∂t= −uj ∂η

∂xj− η

g

∂

∂xj(g uj)

Here, we define η = η′ + η0, f is the Coriolis force and ζ is the relativevorticity. Covariant and contravariant vectors are defined through the short-hand metric tensor notation ui = gijuj , g

ij = (gij)−1. Furthermore, using

εij as the two-dimensional permutation matrix.the divergence and relativevorticity can be calculated as

g ∇ · v =∂

∂xj(g uj), gζ = εij

∂uj∂xi

. (3)

We note that the metric terms can be precalculated and stored once theissue of discretizing the cube has been resolved. To this end, we discuss thediscretization of the cubed-sphere using the spectral element method in section3.1 and the meshless collocation method in section 3.2. An example of theresulting discretized cube is shown in figures (1)

2.1 Semi-Implicit Time Discretization

As an integral part of the hybrid meshless/spectral-element model, the semi-implicit time stepping scheme which we discuss in this subsection has manycomputational advantages. Semi-implicit time-stepping schemes were firstused in atmospheric models in order to aleviate the problem of stability con-straints ultimately due to the fast moving gravity waves in the discrete shal-low water equations [20]. They have been successfully applied for allowing


Fig. 1. Hybrid Cube

an increase in the time step without affecting the atmospherically importantRossby waves. Such a semi-implicit method is described in this subsection andwas originally proposed in the spectral element model developed in [20].

In the hybrid meshless/spectral-element method, the semi-implicit timestepping is composed of an explicit leapfrog scheme for the advection termscombined with a Crank-Nicholson scheme for the gradient and divergenceterms. Adopting the difference notation δui = ui(n+1) − ui(n−1) and δηi =η(n+1) − η(n−1), the time discretized shallow water equations in curvilinearcoordinates can be written as

δui +∆tgij∂

∂xj(δ η) = 2∆t

[− gij ∂

∂xj(η)n−1 + f i(n)

u

],

δηi +∆tη0

g

∂

∂xj(gδuj) = 2∆t

[η0

g

∂

∂xj(guj)n−1 + f i(n)

η

],

where the tendencies fu and fη contain nonlinear terms along with the Coriolisterm, namely

fu = −gij[εjku

k(n)g(f + ζn) +∂

∂xj

(1

2uku

k)n]

,

and

fη = −uj ∂η∂xj

.

Lastly, bringing the implicit terms to the left hand side of the equation andthe explicit terms to the right, we end up with the time discrete evolutionform of the shallow water equations


ui(n+1) +∆tgij∂

∂xj(η)n+1 = ui(n−1) −∆tgij ∂

∂xj(η)n−1 + 2∆tf i(n)

u (4)

ηn+1 +∆tη0

g

∂

∂xj(guj)n+1 = ηn−1 −∆tη0

g

∂

∂xj(guj)n+1 + 2∆tfnη (5)

which is now in the form needed for spatial discretization using the hybridmeshless/spectral-element. Because of the Crank-Nicholson terms, the stor-age of two previous time steps is needed. The overall performance of thissemi-implicit method is reduced to the performance of a robust solver thatcan ultimately be parallelizable in an efficient manner for obtaining the solu-tion at the n + 1 time step. The preconditioned conjugate gradient methodusing a block-jacobi type preconditioner offers such an approach but at thecost of inter-elemental communication at every iteration step of the iterativesolver. The method is of course highly dependent on the spatial approximationscheme and will be discussed in the next two subsections. To this end, we firstbriefly review the construction of the nodal spectral element spproximationon the cubed-sphere.

3 Hybrid Meshless/Spectral-Element Discretization

3.1 Spectral-Element Discretization

The spectral element formulation for the cubed sphere begins by decomposingeach face on the unit cube denoted by Pi, i = 1, . . . , 6 into Ne nonoverlappingsubdomain elements Ωe of equal area which are each referenced to a standardelement Ωst = [−1, 1]2 by a mapping xe(r, s) ∈ Ωe for (r, s) ∈ Ωst. The map-ping also has a well defined inverse (r, s)e(x) ∈ Ωst,x ∈ Ωe. Since two fieldsare being approximated in the discrete shallow water equations, namely thevelocity and the geopotential, two different approximation spaces are needed.As in most shallow water models, we adapt the so-called staggered grid ap-proach to discretization. We begin by defining the approximation space forthe velocity as VN := PN,Ne ∩H1(Pi), where PN,Ne is the space of piecewisecontinuous functions that map to polynomials of degree less than or equal Nto the reference element on face i. Namely,

PN,NE (Pi) := v(xe(r, s))|Ωe ∈ PN (r)⊗ PN (s), e = 1, . . . , Ne,

where PN (r) is the space of all polynomials of degree less than or equal toN . In order to facilitate inter-element continuity on Ωe for all e ∈ [1, Ne]and more globally, inter-face continuity on Pi, i = 1, . . . , 6, nodal Lagrangianinterpolants are used to construct the basis within each element. In this pa-per, we use Lagrangian interpolants constructed from orthogonal Legendrefunctions of degree p.

While many spectral-element and spectral collocation models of fluid dy-namics such as the ones found in [19] have utilized a staggered grid where the


pressure/geopotential field on each element is discretized on an N − 2 Gauss-Legendre distribution which does not include the boundaries of the element,this hybrid meshless/spectral-element model relies on boundary informationof the geopotential field as explained in the next section on the three-fieldformulation for the shallow water equations. We thus build a staggered gridfor the geopotential field which includes the boundaries of each element Ωe

by considering the space MN := P(N−2,Ne)∩L2(Pi) and distributing (N −2)2

Gauss-Lobotto-Legendre points (ξ1i , ξ

2j ).

Using the space MN , the geopotential is expanded in a similar mannerto the velocity components by using the (N − 2)-th degree Lagrangian inter-polants πi as

η(r, s)|Ωe =

N−2∑

i=0

N−2∑

j=0

ηeij πi(r)πj(s),

with ηeij being the nodal values of the geopotential at (ξ1i , ξ

2j ). This expansion

will require evaluation not only on the geopotential grid, but on the velocitygrid as well. As will be shown later, the geopotential field provides the meansfor coupling the spectral element and meshless collocation approximations.

With the definitions of the Lagrangian nodal expansion and quadraturerule in place for each element, we can now apply weak formulation to the semi-implicit time discretized shallow water equations. Find (ui, η) ∈ VN×MN suchthat for all (vi, q) ∈ VN ×MN

〈ui(n+1), vi〉 −∆tgij〈ηn+1,∂

∂xjvi〉 = 〈ui(n−1), vi〉+∆tgij〈ηn−1,

∂

∂xjvi〉+ 2∆t〈f i(n)

u , vi〉

〈ηn+1, q〉+∆tη0

g〈q, ∂

∂xj(guj)n+1〉 = 〈ηn−1, q〉 −∆tη0

g〈q, ∂

∂xj(guj)n+1〉+ 2∆t〈fnη , q〉.

(6)

With the matrices, cubed-sphere metrics, and Coriolis forcing constructedon each element, the semi-implicit scheme can now be formulated into Ne

local Helmholtz problems where the geopotential is solved at every timestepfrom a discrete Helmholtz problem and then ’communicated’ to the velocityfield. Writing the assembled discrete shallow water system from the previoussubsection in matrix-vector form, we get

Bt −Dt

Dt Bt

[

un+1

ηn+1

]=

[Rtu

Rtη

], (7)

where

Bt = B/∆t, Rtu = Ru/∆t, Bt = B/(∆tη0), Rtη = Rη/(∆tη0) (8)

The Helmholtz problem for the geopotential perturbation at each timestep isobtained by solving for the velocity un+1 in the above block system to arriveat


un+1 = B−1(Rtu +∆tg ijDT ηn+1), (9)

and then applying back-substitution to obtain an equation for the geopotential

gBηn+1 +∆t2η0DgB−1g ijDT ηn+1 = R′η, (10)

whereR′η ≡ gRη −∆tη0DgB−1Ru. (11)

Once the geopotential ηn+1 is computed, the velocity components u1, u2 arecomputed from (9) where thereafter, shared local nodal values on elementboundaries of the velocity components are then averaged.

Furthermore, due to the fact that η0, g , B and B−1 are diagonal, andg ij can be shown to be symmetric, it is easy to see that the matrix HSE issymmetric and positive definite. In effect, an efficient preconditioned conju-gate gradient method can be constructed by using local element direct solversfor the Helmholtz problem with zero Neumann pressure gradient boundaryconditions. The inverse of each local Helmholtz operator matrix restricted toan element H|Ωe is computed once and stored for use as a block-Jacobi pre-conditioner. This preconditioning technique enjoys a computational structureideal for parallel processors due to the fact that the preconditioner is strictlylocal to an element and requires no global communication.

3.2 The Empirical Backus-Gilbert Reproducing KernelDiscretization

Coupled with the global spectral-element method for use in regional approxi-mation of the shallow-water model, the empirical Backus-Gilbert reproducingkernel discretization method, originally introduced in Blakely [5] has beendemonstrated to produce highly accurate solutions to time-dependent nonlin-ear PDEs while being endowed with great freedom in choosing the approxima-tion space for building the reproducing kernel. Furthermore, as the name ofthe method suggests, the EBGRK is completely empirical with respect to thedistribution of meshless nodes in the domain of interest. For complete detailsof the method the reader is referred to [5].

The EBGRK method considers a quasi-interpolant of the form

Pu(x) =N∑

i=1

u(xi)Ψi(x). (12)

where u = [u(x1), ..., u(xN )]T represents the given data on a set of N distinctevaluation nodes X = x1, . . . ,xN on a bounded domain Ω ⊂ R2. The finiteset of nodes X is endowed with a seperation distance defined as

qX :=1

2min

xj 6=xi‖xi − xj‖2.


The quasi-interpolant Ψi(x), or discrete reproducing kernel in some literature,is constructed to be minimized in a discrete quadratic expression subject tosome approximation space reproduction constraints. Details on constructingthe empirical reproducing kernel is out of the scope of this paper. We refer thereader to Blakely [5] for the construction of the kernel and efficient numericalimplementation.

To initiate regional meshless approximation on the cubed-sphere, con-sider the domain ΩM = ∪Mi=1Ω

ei constructed of M contiguous elementson the discretized cubed-sphere. For simplicity, we will assume ΩM lies ononly one face of the the cube. Building a meshless approximation spaceVΨ,X begins by randomly distributing two sets of NM distinct collocationnodes in ΩM and on its boundary ∂ΩM giving two sets X VM , X ηM such thatX VM = X ηM . Using the EBGRK method, the kernels Ψi(·) and Φi(·) are con-structed to form the discrete spaces VΨ,X = spanΨi(·), i = 1, . . . , NM andMΦ,X = spanΦi(·), i = 1, . . . , NM and with respect to the sets X VM andX ηM , respectively.

For writing the cubed-sphere shallow water equations in strong form, weutilize the matrix-vector form of the equations originally given in [21]. Asin any other collocation method, we construct the set of Dirac delta testfunctionals ΠXVM = δx1

, . . . , δxNM ⊂ (H1)′(ΩM ) and multiply each of thevelocity components and geopotential by each Dirac delta test functional δxi ∈ΠXVM evaluated which gives

〈δxj , ui(n+1)〉+∆t〈δxj , gij∂

∂xj(η)n+1〉

= 〈δxj , ui(n−1)〉 −∆tgij〈δxj ,∂

∂xj(η)n−1〉+ 2∆t〈f i(n)

u , δxj 〉

〈δxj , ηn+1〉+∆t〈δxj ,η0

g

∂

∂xj(guj)n+1〉

= 〈δxj , ηn−1〉 −∆t〈δxj ,η0

g

∂

∂xj(guj)n+1〉+ 2∆t〈δxj , fnη 〉

(13)

with

〈δxj , f i(n)u 〉 = −εjk〈δxj , gijuk(n)gf〉 − 〈δxj , gijunk

∂

∂xjuk(n)〉,

and

〈δxj , fnη 〉 = −〈δxj , uj∂ηn

∂xj〉.

In order to approximate these equations spatially with the EBGRK method,we look for a solution u ∈ (VΨ,X )2 ⊂ (H1(ΩM ))2 and η ∈ VΨ,X by taking forall n ≥ 0

unk (xj) =

NM∑

i=1

unk (xi)Ψi(xj), ηn(xj) =

NM∑

i=1

ηn(xi)Φi(xj) for xk ∈ X VM ,


where unk (xi) and ηn(xi) are the approximated values at the collocation nodesxi at time step n. Substituting these into (13) and applying the Dirac deltafunctionals, we get for all δxj ∈ ΠXVM

〈δxj ,NM∑

i=1

uk(n+1)(xi)Ψi(x)〉+ 〈∆tgij ∂

∂xjδxj ,

NM∑

i=1

ηn+1(xi)Φi(x)〉

= 〈δxj ,NM∑

i=1

uk(n−1)(xi)Ψi(x)〉+ 〈∆tgij ∂

∂xjδxj ,

NM∑

i=1

ηn−1(xi)Φi(x)〉+ 〈2∆tδxj , f i(n)u 〉

〈δxj ,NM∑

i=1

ηn+1Φi(x)〉+ 〈∆tη0

g

∂

∂xjδxj , g

NM∑

i=1

uj(n+1)(xi)Ψi(x)〉

= 〈δxj ,NM∑

i=1

ηn−1(x)Φi(x)〉 − 〈∆tη0

g

∂

∂xjδxj , g

NM∑

i=1

uj(n−1)(xi)Ψi(x)〉+ 2〈∆tδxj , fnη 〉

(14)

The calculation of

gxj

∂Ψi(xj)

∂xk, xj ∈ X VM

which is used in the divergence terms of the strong form shallow water systemis made prior to time stepping and stored as matrices in the form

Dk =

gx1Υ1(x1) gx1

Υ2(x1) · · · gx1ΥNM (x1)

gx2Υ1(x2) gx2

Υ2(x2) · · · gx2ΥNM (x2)

...gxNM Υ1(xNM ) gxNM Υ2(xNM ) · · · gxNM

ΥN (xNM )

, (15)

where Υj(·) denotes the differential operator ∂∂xk

acting on the kernel Ψj(·),which was shown how to be constructed in section (3.2). A similar matrixgijD

Tk used in calculating the gradient of the geopotential in strong form is

also computed and stored prior to time stepping. These matrices are akinto the two-dimensional derivative matrix Di of size N2 ×N2 in the spectralelement formulation.

Using notation borrowed form the spectral element formulation, we writethe discretized equations in matrix form with D = (D1, D2) as the derivativematrices with respect to the collocation nodes and B, B as the collocationmatrices for the velocity and geopotential, respectively. This leads to the sys-tem

Bt −Dt

Dt Bt

[

un+1

ηn+1

]=

[Rtu

Rtη

],

where


Bt = B/∆t, Rtu = Rt

u/∆t (16)

(17)

Bt = B/(∆tη0), Rtη = Rη/(∆tη0) (18)

Performing the Uzawa velocity-geopotential decoupling algorithm where wesolve for the velocity un+1 in the above block system to arrive at

un+1 = B−1(Rtu +∆tg ijDT ηn+1). (19)

and then applying back-substitution to obtain an equation for the geopotentialat the time step n+ 1.

gBηn+1 +∆t2η0DgB−1g ijDT ηn+1 = R′η (20)

whereR′η ≡ gRη −∆tη0DgB−1Ru.

As with the spectral element case, we are now concerned with the solutionto the Helmholtz problem for the geopotential. Although the meshless col-location method yields a strong form of the discrete shallow water model,the resulting discrete Helmholtz equation is similar to the spectral elementHelmholtz equation in that the matrix operator

HMM = gB +∆t2η0DgB−1g ijDT (21)

is symmetric and positive definite and of size NM × NM . It is important tonotice that the term

DgB−1g ijDT

is a discrete pseudo-Laplacian operator on ΩM , a local domain on the sphereS2. Because of the fact that the domain is locally defined on the sphere, essen-tial boundary conditions on ∂ΩM are needed in order to show direct equiva-lence to a local Helmholtz elliptic problem on ΩM . This was not the case withthe spectral element discretization on the sphere since no boundary conditionsare needed for the global shallow water model. As a result, the boundary in-formation on ∂ΩM must come from the spectral element approximation inorder to produce a unique solution to the discrete Helmholtz problem (20).As we propose in the next section, if ηn+1 is the unique solution to this localHelmholtz problem at time n + 1 with respect to the boundary informationgiven by a global spectral element discretization at time n+1, then ηn+1 is anapproximation to the geopotential field restricted to ΩM at time step n + 1.To accomplish this task, we adapt the three-field domain decomposition algo-rithm developed in [7] for coupling the two Helmholtz discretizations, whichis discussed in the next section.


4 Coupling the meshless and SE approximations

Because the meshless approximation is done locally utilizing the strong for-mulation of the shallow-water equations on a local domain ΩM , certain transi-tion conditions are needed on the boundary of the subdomain connecting themeshless and spectral-element approximations in order to satisfy continuityand flux conditions of the solution along with the artificial fluxes of the fieldvariables. In 1994, Brezzi and Marini (see [7]) developed a method termed thethree-field formulation for hybrid finite-element formulations where the goalwas to give the possibility of coupling different finite-element approximationsusing different meshes and basis functions from one subdomain to another.

In this paper, we extend the idea of the three-field technique to couplespectral-element and meshless collocation methods. As shown in the previoussections, the manner in which we couple the two approximation schemes isdone implicity. Namely, after deriving the semi-implicit method, a symmetricpositive definite discrete Helmholtz type equation was left to be solved ateach time step for the geopotential. With the solution of the geopotential athand, it could then be used to approximate the velocity field at the same timestep. So the question that remains is how to solve Helmholtz equations for thecoupled meshless/spectral-element approximation. In this section we considersolving the elliptic problem

Hu = ∆u(x) + g(x)u(x) = f(x), ∈ S2, (22)

where S2 is the unit sphere which is discretized via the cubed-sphere methoddiscussed in section (2). The fact that spatial discretization is not per-formed with spherical harmonics but rather on a cubed-sphere mesh allowsfor an adaptive localized approximation using meshless collocation via do-main decomposition. Thus the heart of the hybrid shallow water model liesin the efficient handling of the Helmholtz equation on the sphere using themeshless/spectral-element formulation.

For proper stability analysis of this new three-field formulation for couplingspectral-elements and meshless collocation including Babuska-Brezzi inf-suptype conditions, the reader is referred to the paper by Blakely, [6]. In orderto introduce the method, we must first discuss the necessary approximationspaces that will be used in the formulation and discretization and their rele-vant physical meaning.

4.1 The Continuous Three-Field Formulation

Using the notation from the previous subsections, suppose we have are givena subdomain of M unioned elements ΩM = ∪Mi=1Ω

ei . For simplicity of expo-sition of the three-field method, we assume ΩM is on one face of the cube.Let ΓSE/M denote the boundary of ΩM which we will call the interface of thehybrid method between the spectral element and meshless collocation approx-imations. Finally, we denote ΩSE = Ω−ΩM , namely the collection of spectral


elements not in ΩM and then set Ω1 := ΩSE , Ω2 := ΩM and Γi := ∂Ωi, fori = [1, 2], which are the boundaries of these domains sharing the interfaceΓSE/M .

In addition to the Sobolev space H10 (Ωi) on each domain Ωi, utilizing the

interface ΓSE/M leads to two additional types of spaces that will be neededfor domain decomposition. We define a trace space and two dual spaces onΓSE/M by considering H1/2(ΓSE/M ) with corresponding norm ‖ · ‖Γ := ‖ ·‖H1/2(ΓSE/M ) and denote the dual of this space as H−1/2(Γ ). Furthermorewe introduce two spaces of Lagrangian multipliers which provide the role ofenforcing necessary boundary continuity over the interface ΓSE/M and are

defined as Λi := H−1/2(Γi) for i = [1, 2] which can be regarded as the dual ofthe trace spaces associated with the two Hilbert spaces H1(Ω1) and H1(Ω2).The Lagrangian multiplier space is endowed with the standard scalar innerproduct L2(Γ ), 〈Λi, H1/2〉Γ =

∫Γλiu

ids for ui ∈ H1(Ω). The third functionspace which acts as the global continuity space for the hybrid approximationis defined on the interface Γ as restrictions of functions on the sphere S2 tothe interface. Namely

Φ := υ ∈ L2(Γ ) : ∃u ∈ H1(S2), u = υ on Γ. (23)

Global norms for the spaces Vi := H10 (Ωi) and Λi can be given as broken

norms over Ωi

‖u‖V :=( 2∑

i=1

‖ui‖21,Ωi), ‖λ‖Γ :=

( 2∑

i=1

‖λi‖2H−

12

(Γi)),

and can easily be shown to be Hilbert spaces with these induced norms. Fur-thermore, with the use of extension operators, the interface continuity spaceis endowed with the norm

‖ϕ‖Φ := infu|Γ=ϕ

‖u‖1,Ω.

With the three approximation spaces at hand, the three-field formulationof the Helmholtz problem can be written for the two subdomains utilizing theadditional two interface spaces Λi and the global continuity space Φ. Using thedual product notation 〈·, ·〉i = 〈H−1/2(Γi), H

1/2(Γi)〉 the following variationalform is called the three-field formulation. Find u ∈ V, λ ∈ Λ, and ϕ ∈ Φ suchthat

i)∑2i=1

(aΩi(u

i, vi)− 〈λi, vi〉Γi)

=∑2i=1(f, vi)Ωi , ∀v ∈ V,

ii)∑2i=1〈µi, ui − ϕ〉Γi = 0, ∀µ ∈ Λ

iii)∑2i=1〈λi, ψ〉Γi = 0, ∀ψ ∈ Φ

(24)


The bilinear operator aΩi stems from the weak formulation of the Helmholtzequation and is defined as

aΩi(ui, vi) =

∫

Ωi

∇ui∇vi + guividΩi.

Furthermore, the inner products of the form

〈H−1/2(Γi), H1/2(Γi)〉Γi

signify the artificial boundary or interface matching conditions. To be morespecific, the second equation enforces weak continuity along the interface Γiwith the solution ui on Ωi with respect to the interface continuity variable ϕ.The third equation serves two purposes; 1) It further constraines the space ofLagrangian multipliers Λ by adding orthogonality conditions with the inter-face space Φ and 2) It renders the discrete formulation of the above system asa symmetric positive definite system which can then be solved for the globalsolution (u, λ, ϕ) using a preconditioned conjugate gradient method as will beshown in the next subsection.

We first note that a key observation in the three-field formulation comesfrom the first two equations of (24). For a given ϕ on the skeleton Γ , the firsttwo equations are local Dirichlet problems where the boundary conditions onΓi are imposed in the weak sense. Because of this, one can show that the localproblems are well-posed for a given sufficient ϕ. For a complete analysis of thethree field method for coupling meshless and spectral-element approximationsfor elliptic equations, the reader is referred to the paper by Blakely [6]. Thethesis by Rabin [16] and relevent references therein also give much insight tothe three-field variational formulation in the finite-element context.

4.2 Discrete version of the three-field formulation

The difficulty in passing to the discrete formulation the variational problem(24) is in choosing the appropriate discrete subspaces of V, Λ, and Φ. Arbi-trarily choosing the subspaces can lead to unstable solutions of the discretevariational problem primarily due to not satisfying the discrete versions ofthe inf-sup conditions, so careful consideration of the spaces is necessary. Inpast approaches to the method, usually the discretization of the space Vis chosen first and then Λ and Φ are chosen thereafter to satisfy the inf-sup requirements. In this section, we propose a discrete approximation tothe three-fields formulation by considering the spectral-element and meshlesscollocation methods as the discretization tools which will then lead to thehybrid meshless/spectral-element method for the shallow water equations onthe sphere.

With Ω1 defining the domain for the spectral element approximation andthe regional domain Ω2 being allocated for meshless collocation, we definethe space V 1

N := PN,Ne ∩ H1(Ω1), where PN,Ne is the space of piecewise


continuous functions that map to polynomials of degree less than or equal Nto the reference element Ωe. Namely,

PN,E(Ω1) :=v(xe(r))|Ωe ∈ PN (r)⊗PN (s), e = 1, . . . , Ne such that Ωe ∈ Ω1

,

where PN (r) is the space of all polynomials of degree less than or equal toN . To restrict this space to Ω1, we include all Ωe such that Ω1 ∩ Ωe 6= 0.This approximation space will provide each component of the velocity field onthe spectral element partition Ω1. As described in section (3.1), the discretegeopotential space is obtained by utilizing the staggered grid approach andsetting M1

N := PN−2,Ne ∩H1(Ω1). Since the bounderies on each element Ωe

are essential to the three-field method, the N − 2 Gauss-Lobatto-Legendredistribution of nodes is used for the geopotential grid, which is contrary tomany spectral element staggered grids which use Gauss-Legendre nodes forthe geopotential/pressure field.

The regional domain Ω2 allocates a collocation approximation by consider-ing a random (or uniform) distribution of NM distinct collocation nodes ΩM

and on its boundary ∂ΩM giving two sets X VM , X ηM such that X VM = X ηM . Wethen construct the approximation spaceM 2

NM= V 2

NM:= spanΨ1(x), . . . , ΨNM (x)

as defined in section (3.2).With the spaces defined for the velocity and geopotential fields on each

subdomain Ωi, the Lagrangian multiplier spaces Λi for the interface bound-aries Γi can now be constructed by using the spaces M 1

N and M2NM

. Since M1N

defines a spectral approximation of order N−2, we define the Lagrangian mul-tiplier space for Γ1 as the space of Lagrangian interpolants of order less thator equal to N and restricted to Γ1. This is given by

Λ1N = PN,E(Γ1) :=

λ(xe(r))|Ωe ∈ PN−4(r)|Γ1

, e = 1, . . . , Ne such that Ωe∩Γ1 6= 0.

(25)

Using such a space for H−12 (Γ1), it can be shown that the discrete inf-sup

condition for the interface inner product on Γ1 is satisfied. Namely, for someconstant C1,N dependent on the degree N of the spectral elements, we have

infλ1N∈Λ1

N/0sup

η1N∈M1

N/0

〈Bλ1N , η

1N 〉Γ1

‖η1N‖M1

NM

‖λ1N‖Λ1

N

=〈λ1N , η

1N 〉1

‖η1N‖M1

NM

‖λ1N‖Λ1

N

> C1,N

is satisfied. This result is proved in the paper on the Mortar Spectral Elementmethod by Ben Belgacem et al. [3] in a similar interface inner product usingLagrangian multipliers.

In order to complete the space Λ we need the additional interface spaceon Ω2. On the boundary Γ2, a second meshless collocation space for Λ2

N

is constructed using a random distribution of NMΓnodes restricted to the

interface Γ2 producing the set XΓ2. Using the EBGRK method presented in

section (3.2), the Lagrangian multiplier space for Γ2 is taken to be Λ2NM

=


spanΨλ1 (x), . . . , ΨλNM (x) : x ∈ Γ2 ⊂ H−12 (Γ2) where Ψλi (·) denotes the i-th

discrete reproducing kernel function on XΓ2.

Lastly, in order to connect the two pairs of approximation spaces (M 1N , Λ

1N )

and (M2NM

, Λ2NM

) on Ω1 and Ω2, respectively, we build a suitable discrete sub-space of Φ by taking the Lagrangian interpolants constructed from Legendrepolynomials of degree N − 2 restricted to Γ . Namely,

ΦN := ϕ(xe(r))|Ωe ∈ PN−2(r)|Γ , e = 1, . . . , Ne, such that Ωe ∩ ∂Ω 6= 0.

This will ensure that the discrete inf-sup condition for Φ and Λ1N on Γ1 is

satisfied.The last issue we need to resolve in this three field formulation is complying

with the strong form of the shallow water equations on Ω2. To this end,since Ω2 utilizes a meshless collocation technique, we define the set of testdistributions on Ω2 as M2

δ,NM= δxi : xi ∈ XNM

η where δxi is the Diracdelta function at node xi. The original variational formulation in (24) can nowbe modified as follows. Find (η1

N , λ1N , η

2N , λ

2N , ϕ) ∈M1

N⊗Λ1N⊗M2

NM⊗Λ2

NM⊗Φ

such that

i) aΩ1(η1N , χ

1N )− 〈λ1

N , χ1N 〉Γ1

= (f, χ1N )Ω1

, ∀χ1N ∈M1

N ,

ii) 〈µ1N , η

1N − ϕN 〉Γ1

= 0, ∀µ1N ∈ Λ1

N ,

iii) 〈λ1N , ψN 〉Γ1

= 0, ∀ψN ∈ Φ,

(26)

and

i) aΩ2(η2N , χ

2N )− 〈λ2

N , χ2N 〉Γ2

= (f, χ2N )Ω2

, ∀χ2N ∈M2

δ,NM,

ii) 〈µ2N , η

2N − ϕN 〉Γ2

= 0, ∀µ2N ∈ Λ2

NM,

iii) 〈λ2N , ψN 〉Γ1

= 0 ∀ψN ∈ Φ.

(27)

Once the discrete approximation spaces have been chosen and numericalintegration has been done, an efficient manner in solving this is to constructthe Schur compliment system and then apply a conjugate gradient method.To do this, we first write (26) and (27) in algebraic form as:

Aiηi −BTi λi = fi,

−Biηi + CTi ϕ = 0,

Ciλi = 0,

for i = 1, 2. Now applying block Gaussian elimination to the linear system,we obtain a linear system for ϕ as


Sϕ = g, (28)

where S = S1 + S2, g = g1 + g2 and

Si := CiD−1i CTi , gi := CiD

−1i BiA

−1i fi, Di := BiA

−1i BTi , i = 1, 2 (29)

The S matrix can be considered as the Schur compliment matrix with respectto u and λ of the entire system defined above. Furthermore, it was shown byBrezzi in [7] that the Schur compliment S is symmetric and positive definite ifthe matrices BTi and Ci have full rank. One can then apply a conjugate gradi-ent method to the system (28) to obtain the solution of the elliptic problem onthe global domain. It can be remarked that by definition of gi, the calculationof a conjugate gradient iteration requires the solution to the local Helmholtzequation in each subdomain Ωi, i = 1, 2. Block-Jacobi preconditioning is usedto solve each of these local Helmholtz problems by considering zero Neumannconditions for each local problem. This way, each local Helmholtz problemhas a unique solution and in effect, the matrix Ai has an inverse which canbe calculated before time-stepping.

The last issue of the discrete three-field formulation is related to the effi-cient construction of the matrices C and B. As they include the integrationof the basis functions for the Lagrangian multiplier spaces and the interfacespace and are independent of the data, they can be calculated and stored priorto time stepping as well. The matrices have the form

Ci(j, k) = 〈µi,j , φk〉Γ , µi,j ∈ ΛiN , φi,k ∈ ΦN , (30)

Bi(j, k) = 〈ηi,j , µi,k〉Γi , µi,j ∈ ΛiN , ηi,k ∈M iN . (31)

For i = 2, the above calculations involve integration on a spectral gridusing meshless reproducing kernels. The choice of the β parameter and NM

for a given radial basis that constructs the reproducing kernel determines thestability of the entire hybrid model. A study on these parameters is given infull detail in Blakely [6].

5 Numerical Experiments

As is it well established that the spherical rotational shallow-water equationsrepresent a simplified model of the dynamica of the atmosphere, Williamsonet al. [22] have proposed a series of eight test cases for the equations in spher-ical geometry. It is proposed by the authors that in order to have any typeof success with a new numerical scheme for an a climate model, successfullintegrations of the numerical scheme with these test cases are imperative. Thepurpose of the tests are to examen the sensitivites of a numerical scheme withmany computational challenges faced in atmospheric modeling such as sta-bilization of the scheme for large time steps over a long period of time, thepole problem, simulating flows which have discontinuous first-derivatives inthe potential vorticity, and simulating flows over mountain topographies.


5.1 Test Case 2: Global Steady State Nonlinear Zonal GeostrophicFlow

As the second test case, a steady state flow to the full non-linear shallow waterequations is prescribed and the challenge for a numerical scheme is simply totest its numerical stability with respect to l1, linf errors over time. Since theflow is steady, the numerical scheme should be able to integrate the model formany steps without the addition any filtering. The velocity field is given as

u = u0(cos θ cosα+ cosλ sin θ sinα),

v = −u0 sinλ sinα,

which is non-divergent. The analytic geopotential field is give by

η = gh0 −(aΩu0 +

u20

2

)(− cosλ cos θ sinα+ sin θ cosα)2,

with constants u0 = 2πa/(12days) and gh0 = 2.94× 104m2/s2.For this numerical experiment, we began with 24 total spectral elements (4

on each face) and ran the model for 121 days without any additional filtering.A second integration was performed using on face number 2 of the cube themeshless collocation approximation built from compactly supported radialfunctions (see [18]). Figure (2) shows the l1 errors over time of the geopotentialsolution on face 2 for both the spectral element and meshless collocationapproximations.

0

1e-06

2e-06

3e-06

4e-06

5e-06

6e-06

0 20 40 60 80 100 120 140 160

Spectral ElementsMeshless

Fig. 2. Plot of l1 errors of the spectral element and meshless geopotential solutionson face 2 of the cubed sphere using 256 (64 per element) nodes for the evaluation ofthe solutions.

Notice how errors in the meshless approximation do not grow nearly as fastas in the spectral element approximation, despite not being as accurate. This is


due to the collocation properties of the radial basis used. For the geopotentialgrid, a total 256 Gaussian-Lobatto-Legendre nodes (64 per element) wereused at each time step. Furthermore, to obtain an accurate error comparisonbetween the methods, the collocation approximation was evaluated at thespectral element nodes. Similar results for the linf error were also obtained.

5.2 Test Case 6: Rossby-Haurwitz Waves

The most interesting of the test cases features an initial condition for thevelocity which is actually the analytical solution to the non divergent nonlinearBarotropic equation on the sphere, given as a vorticity equation. We referthe reader to [22] for the initial conditions of the velocity components andgeopotential field.

As originally proposed, these waves were expected to evolve nearly steadilywith wavenumber 4. However, Thuburn and Li showed that this case is actu-ally weakly unstable in that it will eventually break down once perturbed. Thisusually occurs after about 40 days depending on the model and parametersused. Figures in (3) show the geopotential layed out on rectangular coordi-nates for easier viewing. The figure to the left shows the field after 10 daysat an angle where the hybrid mesh structure on the cubed-sphere can easilybe seen. Notice that the continuity along the interfaces between the mesh-less and spectral-element approximations are preserved, meaning the inf-supconditions along the interface are satisfied.

6 Conclusion

In this article, we proposed and developed a new hybrid numerical schemefor the shallow water equations on the sphere based on the merging of sev-eral numerical tools including meshless collocation, spectral elements, and thethree-field variational formulation. Furthermore, a high-performance Fortran90 software suite has been developed for the hybrid method for use on dis-tributed memory parallel processors with the message passing interface. Sucha successful high-performance implementation ultimately required the use ofother Fortran 90 numerical packages for almost half of the computational tasksin the model, such as the domain decomposition. Although much work stillremains with theoretical issues of the hybrid approximation scheme such asstability and convergence, the numerical examples in the previous section haveclearly shown the method’s robustness in approximating the global solutionwith spectral elements along with localized regions using meshless collocation.

Acknowledgments. The author would like to thank Franco Brezzi for manyfrutiful discussions about the three-field method implementation and stabilityresults, and for proof-reading much of the paper. I would also like to thank myadvisors Profs. John Osborn and Ferdinand Baer for aiding me in the theoret-ical issues behind the shallow-water equations and mixed/hybrid methods.


Fig. 3. Plots of geopotential approximation using 20 spectral elements and 4 ele-ments allocated to meshless collocation. Plot after 10 and 60 days

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